Find the amount of each payment to be made into a sinking fund so that enough will be present to accumulate the following amount. Payments are made at the end of each period: $5,700; money earns 3.6% compounded monthly for 5 1/4 years.
Set up the formula to find the amount of each payment. Select the correct choice and fill in the answer boxes to complete your choice. (Type integers or decimals. Simplify your answers.)
A. PMT = (1+ )
B. PMT = [(1+)^-1]
C.
Answer (2)
The amount of each payment to be made into the sinking fund is approximately $82.33. This is calculated using the future value of the sinking fund formula and taking into account the interest rate and payment periods. The process involves calculating the monthly interest rate and the total number of payments needed to reach the future value of $5,700.
;
Calculate the monthly interest rate: i = 12 0.036 = 0.003 .
Calculate the total number of payments: n = 5.25 × 12 = 63 .
Use the sinking fund formula to find the payment amount: PMT = 5700 × ( 1 + 0.003 ) 63 − 1 0.003 .
Simplify the expression to find the PMT: PMT ≈ 82.33 .
Explanation
Problem Analysis We are asked to find the amount of each payment to be made into a sinking fund. We are given the future value of the sinking fund, the interest rate, the compounding period, and the term of the sinking fund.
Given Information The future value of the sinking fund is $FV = 5 , 700 . The annual interest rate is 3.6% , compounded monthly. The term of the sinking fund is 5 4 1 = 5.25 years. Payments are made at the end of each month.
Sinking Fund Formula We need to find the amount of each payment (PMT) to be made into the sinking fund. The formula for the future value of an ordinary annuity (sinking fund) is:
F V = PMT ∗ i ( 1 + i ) n − 1
Where:
FV is the future value of the sinking fund
PMT is the payment amount
i is the interest rate per period
n is the number of periods
Rearrange the Formula We can rearrange the formula to solve for PMT:
PMT = F V ∗ ( 1 + i ) n − 1 i
Calculate i and n First, we need to calculate the monthly interest rate (i) and the total number of payments (n):
i = 12 0.036 = 0.003
n = 5.25 ∗ 12 = 63
Substitute Values Now, we can substitute the values into the formula:
PMT = 5700 ∗ ( 1 + 0.003 ) 63 − 1 0.003
PMT = 5700 ∗ ( 1.003 ) 63 − 1 0.003
Calculate PMT Now, we calculate ( 1.003 ) 63 :
( 1.003 ) 63 ≈ 1.20867
So, the equation becomes:
PMT = 5700 ∗ 1.20867 − 1 0.003
PMT = 5700 ∗ 0.20867 0.003
PMT = 5700 ∗ 0.01437
PMT ≈ 82.33
Final Answer Therefore, the amount of each payment to be made into the sinking fund is approximately $82.33 .
Examples
Sinking funds are commonly used by businesses and individuals to save up for a future expense or debt. For example, a company might use a sinking fund to save money each month to prepare for a large debt payment that is due in several years. An individual might use a sinking fund to save for a down payment on a house or a new car. By making regular payments into a sinking fund, the company or individual can ensure that they will have enough money to cover the expense when it comes due. The sinking fund ensures a systematic approach to saving, turning small, regular contributions into a substantial sum over time, aided by the power of compound interest.
